Morita equivalence and Morita duality for rings with local units and subcategory of projective unitary modules

TL;DR

This paper extends Morita theory and duality concepts to rings with local units, broadening classical results to a more general setting involving projective and injective modules.

Contribution

It generalizes Morita equivalence and Azumaya-Morita duality theorems to rings with local units using categories of finitely generated projective and injective modules.

Findings

Extended Morita equivalence to rings with local units.

Generalized Azumaya-Morita duality theorem for such rings.

Provided new categorical characterizations of module categories.

Abstract

We study Morita equivalence and Morita duality for rings with local units. We extend the Auslander's results on the theory of Morita equivalence and the Azumaya-Morita duality theorem to rings with local units. As a consequence, we give a version of Morita theorem and Azumaya-Morita duality theorem over rings with local units in terms of their full subcategory of finitely generated projective unitary modules and full subcategory of finitely generated injective unitary modules.